Abstract: This paper explores the potential of Lagrangian duality for learning applications that feature complex constraints. Such constraints arise in many science and engineering domains, where the task amounts to learning to predict solutions for constraint optimization problems which must be solved repeatedly and include hard physical and operational constraints. The paper also considers applications where the learning task must enforce constraints on the predictor itself, either because they are natural properties of the function to learn or because it is desirable from a societal standpoint to impose them.